Not unrealistic, take a look at Hal Higdons 15k/10 mile novice training plan which just happens to be 10 weeks. Plenty of other plans on the web too, have a search round and find something you like the look of and can fit your schedule.

It's possible but if you keep pushing it like that you're likely to end up injured and unable to run. It's great that you want to continue running but the body needs time to adapt.
If you go for it good luck, add a mile a week to your long run with a couple of cut back weeks and 2 weeks taper should get you through, but my recommendation would be to give yourself more time to build up to half marathon distance.

Have you been doing this work for a long time? If not you might need more time to adapt to it before your ready for regular running. At least your getting a workout at work rather than sitting down all day but that won't help with your half marathon plans (well actually it might but only if you can get in the runs too without wiping yourself out).

Hi, I'm currently working similar shifts after many years of office type work.

When I first started this type of work I didn't run at all but after a few months when it didn't take so much out of me I started to reintroduce running. What I try to do now is the majority of my running on days off and then run every other day during the working week. I've had lots of job and hence shift changes in the last year and had to adapt to changing shifts but now that I'm more used to it I love being able to go out for a run when most people are at work.

If I'm starting lates then running is easy, I have a lie in then go for a run then home for second breakfast/ lunch before starting work. If working early then it's harder but I go out for a run as soon as I get in or sometimes stop on the way home for a run as a change of scenery, if I leave it too long after finishing work then the run won't happen. What really wipes me out is working nights but I can still get a run in after work and before laying awake most of the day trying to sleep.

It sounds like your work is Monday to Friday with Saturday and Sunday off? If so I would be aiming to run Saturday and Sunday one of which would be your long run then just Tuesday and Thursday during the week.

I get what your saying about the difficulty of sleeping when on changing shifts but I feel that I get better sleep now that I'm running regularly again than when I wasn't.

Good luck, I hope you manage to find something that works for you to achieve your half marathon goals.

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